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**|**Biomed Opt Express**|**v.2(8); 2011 August 1**|**PMC3149533

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- Abstract
- 1. Introduction
- 2. Optical Torques over NRI Spherical Particles
- 3. Numerical Results and Discussion
- 4. Conclusions
- References and links

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Biomed Opt Express. 2011 August 1; 2(8): 2354–2363.

Published online 2011 July 22. doi: 10.1364/BOE.2.002354

PMCID: PMC3149533

Received 2011 May 2; Revised 2011 July 21; Accepted 2011 July 22.

Copyright ©2011 Optical Society
of America

This is an open-access article distributed under the terms of the Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 Unported License, which permits download and redistribution, provided that the original work is properly cited. This license restricts the article from being modified or used commercially.

We investigate optical torques over absorbent negative refractive index spherical scatterers under the influence of linear and circularly polarized TEM_{00} focused Gaussian beams, in the framework of the generalized Lorenz-Mie theory with the integral localized approximation. The fundamental differences between optical torques due to spin angular momentum transfer in positive and negative refractive index optical trapping are outlined, revealing the effect of the Mie scattering coefficients in one of the most fundamental properties in optical trapping systems.

Since 2000, when the first papers began to appear treating the subject of constructing some artificial medium with simultaneous negative permittivity and permeability [1,2], there has been an increasing interest on the new properties and revolutionary potential applications of what has been called negative refractive index (NRI) or double-negative (DNG) metamaterials, left-handed (LH) materials or Veselago’s medium (VM), which are artificial structural-arranged materials capable of delivering a homogeneous medium with an effective negative refractive index [3–5]. Some of these applications overcome current positive refractive index (PRI) limitations and, together with plasmonic structures, they are promising near-future technologies and devices for both microwaves and optics, such as in lenses, transmission lines, antennas, optical cloaking, cancer treatment and so on [6–14].

Recently, we have proposed the use of NRI metamaterials in optical trapping systems not only as optical devices for mechanical and lasing purposes, but as real trappable micro- or nano-particles. We have called this a “double-negative optical trapping” [15,16] or, alternatively, “negative refractive index optical trapping” [17]. Radial and axial radiation pressure forces were then calculated for both Gaussian and Bessel beams using first a ray optics approach, and further the generalized Lorenz-Mie theory (GLMT) with the integral localized approximation (ILA), thus allowing an all-optical regime analysis [16–19]. New and interesting trapping characteristics which could never be observed for any PRI particle were revealed for homogeneous and lossless simple NRI spheres.

This paper extends our previous works and shows how the polarization state (linear or circular) of a TEM_{00} focused Gaussian beam affects the optical torque exerted on lossy (absorbent) NRI spherical particles in comparison to conventional PRI particles of the same geometry. This kind of optical torque arises from the spin angular momentum (SAM) of the incident photons, in contrast to the torque produced by orbital angular momentum (OAM) transfer due to azimuthally asymmetric intensity profiles of incident beams such as Laguerre-Gaussian or high-order Bessel beams [20–22]. Due to the new resonances observed on the Mie scattering coefficients, every optical property (scattered and internal fields, forces, torques, scattering, extinction and absorbing cross sections and so on) is eventually affected in its amplitude or phase when the real part of the refractive index of the particle changes sign, the condition *n _{rel}* = −1 playing no significant rule as does

An arbitrary electromagnetic wave can carry both spin and orbital angular momentum, the first being associated with its state of polarization, and the second with its azimuth light pattern dependence. The mechanism by which both are transferred to a given object is well established. It is well known, for example, that off-axis particles may rotate under the influence of a plane-polarized (linearly polarized) TEM_{00} focused Gaussian beam due to an asymmetric linear momentum transfer [20,23]. Also, a circularly polarized laser beam carries an SAM of *σħ*/photon, where *σ* = 0, + 1 or −1 for linear, right- and left-hand polarizations, respectively, and this AM causes a particle to rotate about its own axis, the sense of rotation being determined by the state of polarization of the beam [20]. Finally, OAM of *lħ*/photon is also carried by the fields, *l* being the topological charge or the mode index. A Laguerre-Gaussian beam, for example, has an AM of (*σ* + *l*)*ħ*/photon [21,22].

Regardless of the kind of AM, Polaert *et al.* have shown that, in the framework of the GLMT, which is an extension of the Mie theory for plane waves [23,24], the Cartesian components of the optical torques exerted on a given object, whose centre coincides with the origin of the rectangular coordinate system, are given by [23]:

$${T}_{x}=-\frac{2M}{c}\frac{\pi}{{k}^{3}}{\displaystyle \sum _{p=1}^{\infty}{\displaystyle \sum _{n=p}^{\infty}\frac{2n+1}{n\left(n+1\right)}\frac{\left(n+p\right)!}{\left(n-p\right)!}\mathrm{Re}\{\begin{array}{l}({g}_{n,TM}^{p-1}{g}_{n,TM}^{p,}[2{\left|{a}_{n}\right|}^{2}-({a}_{n}+{a}_{n}^{})]+({g}_{n,TE}^{p-1}{g}_{n,TE}^{p,}[2{\left|{b}_{n}\right|}^{2}-({b}_{n}+{b}_{n}^{})]\}\\ ,\end{array}}}$$

(1)

$${T}_{y}=-\frac{2M}{c}\frac{\pi}{{k}^{3}}{\displaystyle \sum _{p=1}^{\infty}{\displaystyle \sum _{n=p}^{\infty}\frac{2n+1}{n\left(n+1\right)}\frac{\left(n+p\right)!}{\left(n-p\right)!}\mathrm{Im}\{\begin{array}{l}({g}_{n,TM}^{p-1}{g}_{n,TM}^{p,}[2{\left|{a}_{n}\right|}^{2}-({a}_{n}+{a}_{n}^{})]+({g}_{n,TE}^{p-1}{g}_{n,TE}^{p,}[2{\left|{b}_{n}\right|}^{2}-({b}_{n}+{b}_{n}^{})]\}\\ ,\end{array}}}$$

(2)

$${T}_{z}=-\frac{4M}{c}\frac{\pi}{{k}^{3}}{\displaystyle \sum _{p=-\infty}^{\infty}{\displaystyle \sum _{n=\left|p\ne 0\right|}^{\infty}p\frac{2n+1}{n\left(n+1\right)}\frac{\left(n+\left|p\right|\right)!}{\left(n-\left|p\right|\right)!}\left[\begin{array}{l}{\left|{g}_{n,TM}^{p}\right|}^{2}\left(\mathrm{Re}\left({a}_{n}\right)-{\left|{a}_{n}\right|}^{2}\right)+\\ {\left|{g}_{n,TE}^{p}\right|}^{2}\left(\mathrm{Re}\left({b}_{n}\right)-{\left|{b}_{n}\right|}^{2}\right)\end{array}\right]}},$$

(3)

where *c* is the speed of light in vacuum, *M* the refractive index of the medium surrounding the particle, *k* the wavenumber of the incident light, *a _{n}* and

Consider now that the scatterer is a homogeneous sphere with refractive index *N* = *N _{re}* –

In the next section, we show the behavior of the optical torque components for a plane-polarized (*x*-polarized) focused Gaussian beam with the particle being transversally displaced from the optical axis. Then, we observe the longitudinal *z*-component of the torque produced by circularly polarized beams due to *σ* ≠ 0, a situation that allows us to trap and rotate a NRI particle in a similar manner as currently performed with PRI microparticles.

When a linearly polarized TEM_{00} laser beam hits a PRI homogeneous spherical particle, no torque is observed if the particle is located at the trap focus or, equivalently, at the point of stable equilibrium, regardless of its refractive index being real or complex. However, if the particle is transversally shifted along the trapping plane (perpendicular to the optical axis of the beam and containing the trap focus), it is well-known that an optical torque can be detected, due to an asymmetrical illumination, whenever this particle has a nonzero imaginary refractive index different from that of the external medium [20,23,30]. This *N _{im}* ≠ 0 condition is also valid in the NRI case for achieving nonzero optical torques, as we shall see.

To observe this angular momentum transfer, suppose an *x*-polarized Gaussian beam with wavelength *λ* = 384 nm and beam waist radius *ω*_{0} = 3.7 μm incident on an absorbent spherical particle with *N _{im}* = 10

By looking at Fig. 1, it can be inferred that, for the parameters chosen, the magnitude of *T _{x}* for a NRI particle resemble that of the equivalent PRI particle. But significant differences in magnitude can also be expected for specific values of

Notice that, because of the ray optics characteristic *a*/*λ* ≈19.53, a significant number of Mie coefficients are necessary to account for a good description of the optical torque components. Although the MSCs can assume different complex values for PRI and NRI particles, it is only when *a* ~*λ* or *a* << *λ* that *T _{x}* for a NRI sphere presents a distinct amplitude profile relative to that of a PRI sphere. Figure 2
is a plot of

Finally, Figs. 3(a)
and 3(c) show three-dimensional views of *T _{x}* as both

In an optical tweezers system employing focused Gaussian beams for optically trapping biological molecules and PRI particles in general, SAM is transferred from the incident photons to an absorbing particle located at the trap focus leading to a nonzero longitudinal torque *T _{z}* [20]. This causes the trapped particle to rotate counter- or clockwise along

Although the previous analysis for linear polarized light reveals different radial torque profiles and amplitudes for particular values of *N _{r}* for PRI and NRI particles, it is hard to observe such torques in a real experiment, basically because even NRI particles would necessarily be attracted towards the beam waist centre or repelled away from it [17]. In this way,

Let us again assume a first-order Davis description for a right-hand circularly polarized focused Gaussian beam with *λ* = 384 nm, *ω*_{0} = 3.7 μm propagating along + *z*. The particle is displaced along *z* and (*x*_{0},*y*_{0}) = (0,0) with parameters *a* = 7.5 μm and *N _{im}* = 10

From Figs. 4(a) and 2(c), one concludes that no significant differences in *T _{z}* can be observed for the NRI and PRI curves using the adopted parameters, even though

Figure 6
is a three-dimensional view of *T _{z}* as the complex refractive index of the particle is changed. The right-hand circularly polarized Gaussian beam has the same wavelength and beam waist as before, while the radius of the scatterer is fixed at

The peak amplitudes in Fig. 2 and still valid for *T _{z}* are essentially a consequence of a combination of the first ten

The study of optical torques in optical trapping systems is extremely important and serves as an useful theoretical tool for predicting whether some biological particle will rotate, about some specific axis, under the presence of some arbitrary incident beam. In this way, the GLMT is an essential mathematical formulation to account for numerical optical torque calculations because it can be used to describe the linear or angular momentum transfer from any laser beam to an arbitrary particle in any optical regime. The integral localized approximation reduces computational time in the sense that it eliminates the undesirable and time consuming quadratures with double or triple integration.

The Mie scattering coefficients present different phases and amplitudes depending on the geometry and the electromagnetic properties of the scatterer, and this is also true when we suppose an absorbing negative refractive index spherical particle. In this situation, new resonances appear which reflects our results for optical torques due to the polarization of the incident beam. The inclusion of losses serves to make our model for the NRI particle more physical, and is fundamental in SAM transfer analysis. But we do may expect that our previous analysis for lossless particles still applies for NRI spherical scatterers with very low losses (either due to the dispersive nature of these metamaterials or because of gain) at the operating frequency of the laser beam.

The focused Gaussian beams explored here are not capable of transferring orbital angular momentum due to its azimuth symmetry, and we can naturally expect that other types of laser beams such as Laguerre-Gaussian and higher order Bessel beams, for example, will induce new optical torques in NRI particles.

Experimental verification of our results is still a challenge because of actual technological limitations in nanofabricating effective homogeneous negative refractive index spherical particles, especially in the optical regime. Although this may seem a little frustrating, it would be possible, in principle, to design a delicate experiment using macrostructures with a 2D NRI response in microwaves, small enough and with such a mass that, when impinged by a well-designed laser beam with sufficient power, it is mechanically oriented in a given plane as predicted by our recent studies.

The authors wish to thank FAPESP – Fundação de Amparo à Pesquisa do Estado de São Paulo – under contracts 2009/54494-9 (L. A. Ambrosio’s post doctorate grant) and CePOF, Optics and Photonics Research Center, 2005/51689-2 for supporting this work.

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